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Theorem cfinufil 23952
Description: An ultrafilter is free iff it contains the Fréchet filter cfinfil 23917 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
cfinufil (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ Fin} ⊆ 𝐹))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑋

Proof of Theorem cfinufil
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elpwi 4612 . . . . 5 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
2 ufilb 23930 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
32adantr 480 . . . . . . . . 9 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
4 ufilfil 23928 . . . . . . . . . . . 12 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
54adantr 480 . . . . . . . . . . 11 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝐹 ∈ (Fil‘𝑋))
6 filfinnfr 23901 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑋𝑥) ∈ 𝐹 ∧ (𝑋𝑥) ∈ Fin) → 𝐹 ≠ ∅)
763exp 1118 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → ((𝑋𝑥) ∈ 𝐹 → ((𝑋𝑥) ∈ Fin → 𝐹 ≠ ∅)))
87com23 86 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘𝑋) → ((𝑋𝑥) ∈ Fin → ((𝑋𝑥) ∈ 𝐹 𝐹 ≠ ∅)))
95, 8syl 17 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ Fin → ((𝑋𝑥) ∈ 𝐹 𝐹 ≠ ∅)))
109imp 406 . . . . . . . . 9 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → ((𝑋𝑥) ∈ 𝐹 𝐹 ≠ ∅))
113, 10sylbid 240 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → (¬ 𝑥𝐹 𝐹 ≠ ∅))
1211necon4bd 2958 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → ( 𝐹 = ∅ → 𝑥𝐹))
1312ex 412 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ Fin → ( 𝐹 = ∅ → 𝑥𝐹)))
1413com23 86 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ( 𝐹 = ∅ → ((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
151, 14sylan2 593 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → ( 𝐹 = ∅ → ((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
1615ralrimdva 3152 . . 3 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ → ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
174adantr 480 . . . . . . . . . . . 12 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → 𝐹 ∈ (Fil‘𝑋))
18 uffixsn 23949 . . . . . . . . . . . 12 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → {𝑦} ∈ 𝐹)
19 filelss 23876 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ {𝑦} ∈ 𝐹) → {𝑦} ⊆ 𝑋)
2017, 18, 19syl2anc 584 . . . . . . . . . . 11 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → {𝑦} ⊆ 𝑋)
21 dfss4 4275 . . . . . . . . . . 11 ({𝑦} ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
2220, 21sylib 218 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
23 snfi 9082 . . . . . . . . . 10 {𝑦} ∈ Fin
2422, 23eqeltrdi 2847 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin)
25 difss 4146 . . . . . . . . . . 11 (𝑋 ∖ {𝑦}) ⊆ 𝑋
26 filtop 23879 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
27 elpw2g 5339 . . . . . . . . . . . 12 (𝑋𝐹 → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
2817, 26, 273syl 18 . . . . . . . . . . 11 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
2925, 28mpbiri 258 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋)
30 difeq2 4130 . . . . . . . . . . . . 13 (𝑥 = (𝑋 ∖ {𝑦}) → (𝑋𝑥) = (𝑋 ∖ (𝑋 ∖ {𝑦})))
3130eleq1d 2824 . . . . . . . . . . . 12 (𝑥 = (𝑋 ∖ {𝑦}) → ((𝑋𝑥) ∈ Fin ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin))
32 eleq1 2827 . . . . . . . . . . . 12 (𝑥 = (𝑋 ∖ {𝑦}) → (𝑥𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
3331, 32imbi12d 344 . . . . . . . . . . 11 (𝑥 = (𝑋 ∖ {𝑦}) → (((𝑋𝑥) ∈ Fin → 𝑥𝐹) ↔ ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
3433rspcv 3618 . . . . . . . . . 10 ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
3529, 34syl 17 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
3624, 35mpid 44 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝐹))
37 ufilb 23930 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑦} ⊆ 𝑋) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
3820, 37syldan 591 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
3918pm2.24d 151 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (¬ {𝑦} ∈ 𝐹 → ¬ 𝑦 𝐹))
4038, 39sylbird 260 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝐹 → ¬ 𝑦 𝐹))
4136, 40syld 47 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → ¬ 𝑦 𝐹))
4241impancom 451 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)) → (𝑦 𝐹 → ¬ 𝑦 𝐹))
4342pm2.01d 190 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)) → ¬ 𝑦 𝐹)
4443eq0rdv 4413 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)) → 𝐹 = ∅)
4544ex 412 . . 3 (𝐹 ∈ (UFil‘𝑋) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → 𝐹 = ∅))
4616, 45impbid 212 . 2 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
47 rabss 4082 . 2 ({𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ Fin} ⊆ 𝐹 ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹))
4846, 47bitr4di 289 1 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ Fin} ⊆ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  {crab 3433  cdif 3960  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   cint 4951  cfv 6563  Fincfn 8984  Filcfil 23869  UFilcufil 23923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1o 8505  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fbas 21379  df-fg 21380  df-fil 23870  df-ufil 23925
This theorem is referenced by: (None)
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